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Shape descriptors have recently gained popularity in shape matching, statistical shape modeling, etc. Their discriminative ability and efficiency play a decisive role in these tasks. In this paper, we first propose a novel handcrafted anisotropic spectral descriptor using Chebyshev polynomials, called the anisotropic Chebyshev descriptor (ACD); it can effectively capture shape features in multiple directions. The ACD inherits many good characteristics of spectral descriptors, such as being intrinsic, robust to changes in surface discretization, etc. Furthermore, due to the orthogonality of Chebyshev polynomials, the ACD is compact and can disambiguate intrinsic symmetry since several directions are considered. To improve the ACD’s discrimination ability, we construct a Chebyshev spectral manifold convolutional neural network (CSMCNN) that optimizes the ACD and produces a learned ACD. Our experimental results show that the ACD outperforms existing state-of-the-art handcrafted descriptors. The combination of the ACD and the CSMCNN is better than other state-of-the-art learned descriptors in terms of discrimination, efficiency, and robustness to changes in shape resolution and discretization.

RBF-based methods play a very important role in the point cloud reconstruction field. However, solving a linear system is the bottleneck of such methods, especially when there are a large number of points and lead the computing to be time-consuming and unstable. In this paper, we firstly construct a novel implicit progressive-iterative approximation framework based on RBFs, which could elegantly reconstruct curves and surfaces or even higher dimensional data in an approximation or interpolation way, avoiding expensive computational cost on solving linear systems. Then, we further accelerate the proposed method with a strategy inspired from the conjugate gradient algorithm. In our framework, using proper RBFs allows to simply transform the iteration matrix to be symmetrical and positive definite. Such a property contributes to reduce the computational cost greatly and produce high-quality reconstruction results. Plenty of numerical examples on various challenging data are provided to demonstrate our efficiency, effectiveness, and superiority to other methods.

In this work, we deal with the region control of $$C^2$$ interpolation curves and surfaces using a class of rational interpolation splines in one and two dimensions. Simple sufficient data-dependent constraints are derived on the local control parameters to generate $$C^2$$ interpolation curves lying strictly between two given piecewise linear curves and $$C^2$$ interpolation surfaces lying strictly between two given piecewise bi-liner blending quintic interpolation surfaces. Moreover, we also develop an algorithm concerning the application of the $$C^2$$ rational interpolation spline surfaces on image interpolation.

Metal–organic frameworks (MOFs) are a new type of porous crystal materials that have been widely used in various fields. Owing to their adjustable porous structure, MOF photocatalysts with high catalytic activity have become a research hotspot. In this study, NH2-MIL-125/CoSx samples were investigated as a novel visible light-responsive catalyst for the removal of organic dye. The NH2-MIL-125/CoSx-20 composites showed the best photodegradation efficiency, where photodegradation of the rhodamine B reached 95.4% in 45 min. CoSx was able to effectively extend the wavelength absorption range of NH2-MIL-125 and effectively separate the photogenerated electrons and holes and that could lead to the improved photocatalytic performance of the composites. The reactive superoxide radical (·O2−) was identified as the main active species using reactive species capture experiments, and a reasonable photocatalytic mechanism was proposed. This paper is intended to guide the development of organic–inorganic hybrid materials for environmental remediation.

Tooth surface model is a fundamental element used in the design, analysis, and manufacturing of spiral bevel gears. Based on the face-milled generated process, the tooth surface of spiral bevel gears is mathematically calculated as a complicated representation related to cutter parameters and machine settings. With the representation, the discretized points and curves can be easily obtained from the parameterization of tooth surface. However, the discretized curves for the face-milled generated spiral bevel gears are usually too non-equidistant to directly generate 3D (three dimensional) and mesh models in commercial softwares. This paper presents a novel method to recalculate tooth surface points to generate new approximately equidistant points and curves. The tooth surface representation is introduced at the beginning. Subsequently, a mapping between the isoparametric curve and new approximately equidistant curve is used to recalculate the tooth surface points. To solve the nonlinear equation established from the mapping, a new global optimization algorithm, named real-coded relaxation genetic algorithm (RRGA), is proposed to stably obtain the solutions. Examples are given to validate the proposed method.

This paper presents a method for generating implicit surfaces from polygon soups based on compactly supported radial basis functions (CSRBFs). The surface is represented as the zero level set of an implicit function which interpolates the polygonal data with their outward normal constraints. By specifying two parameters, the support size and the shape parameter, users can flexibly control the accuracy of the reconstructed surfaces. For determining coefficients of RBFs, our method uses a quasi-interpolation framework to avoid solving a large linear system, which allows processing large meshes efficiently and robustly. Moreover, a relationship between the shape parameter and the support radius is provided for the quasi-solution validity, and an error bound of the reconstructed surfaces approximating the original models is deduced through the rigorous theoretical analysis.

At present, it is insufficient to understand the basic data characteristics of the correlated X-ray scattering. And there is a great challenge about how to master the nature of the data. So it is difficult to use and analyze the experimental data more effectively. In addition, there are many reasons, for the experimental artifacts such as whether the shutter is on or off, whether there is the beam line or not, the swaying of the nozzle and the shadow of the detector. So it is rather challenging to analyze the scattering patterns. The purpose of this paper was to develop a method to filter the invalid scattering data and provide the theoretical and experiment fundamentals for studying the X-ray scattering data of the complex biological sample further. The helium molecules were scattered by the X-ray free-electron laser in Spring8 in Japan. And millions of scattering patterns were obtained from the X-ray free-electron laser experiment. Through the analysis of the scattering data, the sum, mean, median and variance of the scattering intensity were obtained. Then different clusters were obtained with the density-based spatial clustering of applications with noise (DBSCAN) algorithm. Based on the DBSCAN, some of the scattering patterns with high artifacts were removed and different clusters were clarified. So the experimental scattering data could be analyzed more effectively. The theoretical and experiment fundamentals for comprehensively studying the X-ray scattering data of the complex biological sample were provided. After the data filtering, the angular autocorrelation of different clusters with Kam’s method will be computed and analyzed effectively.

Based on the Hermite variational implicit surface reconstruction presented in Pan et al. (Science in China Series F: Information Sciences 52(2):308---315, 2009), we propose a multi-level interpolation method to overcome the problems resulted from using compactly supported radial basis functions (CSRBFs). In addition, we present a multi-level quasi-interpolation method which directly uses normal vectors to construct non-zero constraints and avoids solving any linear system, a common step of variational surface reconstruction, and leads to a fast and stable surface reconstruction from scattered points. With adaptive support size, our approach is robust and can successfully reconstruct surfaces on non-uniform and noisy point sets. Moreover, as the computation of quasi-interpolation is independent for each point, it can be easily parallelized on multi-core CPUs.

Convolution surfaces are attractive for modeling objects of complex evolving topology. This paper presents some novel analytical convolution solutions for planar polygon skeletons with both finite-support and infinite-support kernel functions. We convert the double integral over a planar polygon into a simple integral along the contour of the polygon based on Green’s theorem, which reduces the computational cost and allows for efficient parallel computation on the GPU. For finite support kernel functions, a skeleton clipping algorithm is presented to compute the valid skeletons. The analytical solutions are integrated into a prototype modeling system on the GPU (Graphics Processing Unit). Our modeling system supports point, polyline and planar polygon skeletons. Complex objects with arbitrary genus can be modeled easily in an interactive way. Resulting convolution surfaces with high quality are rendered with interactive ray casting.

Tolerance design has a great impact on the cost and quality of a product. Previous research focused on process tolerances or robust tolerance design with little consideration on real manufacturing context. This paper presents a nonlinear method for robust tolerance design based on the real manufacturing context in three stages. The objective function to be minimized is the total manufacturing cost. The constraint equations for the optimization model are also deduced, which select suitable manufacturing processes based on the manufacturing environment. Simulation annealing (SA) is used for the nonlinear optimization. The approach is finally illustrated by a practical example. The results of the comparison with different models indicate that the proposed approach is more effective with the manufacturing resource. The robust and reliable tolerance can be obtained.